Predictable Quotes and Inventory Risk in Market-Making Models

Summary

The document explains the admissible quote controls in an exponential-utility market-making objective. A market maker chooses ask and bid quote offsets to maximize expected utility of terminal cash plus the value of terminal inventory at the asset price. The admissible controls are predictable, meaning quote decisions may use information available at the time but cannot rely on future information. The source also describes the controls as bounded from below, a technical restriction used to support the mathematical results.

In the paper’s trading interpretation, the market maker observes order flow and the inventory changes it produces, then adjusts quotes accordingly. The response contrasts this information set with models that allow some participants to act with future information, such as informed asset managers responding to news. The explanation clarifies the optimization and information assumptions, but does not derive the optimal quote strategy or specify the model’s full market dynamics.

Key ideas

• The objective maximizes expected exponential utility of terminal cash and marked-to-market inventory.
• Predictable quote controls use information available when decisions are made and exclude future information.
• The quote controls are bounded from below as a technical assumption for the mathematical results.
• In the described setup, a market maker observes order flow and inventory when adjusting quotes.

Tags

# Dealing with Inventory Risk


# Dealing with Inventory Risk












I am reading a paper$^\color{magenta}{\dagger}$ on market-making and having trouble understanding a point. Towards the end of section 2, the authors stated that:

$$\sup_{(\delta_t^a)_t, (\delta_t^b)_t \in \mathcal A} {\Bbb E} \left[ - \exp \left( - \gamma \left( X_T + q_T S_T \right) \right) \right]$$

where $\mathcal A$ is the set of predictable process bounded from below. Can someone please explain what this means - in the mathematical context and in the trading context for a trader?

$\color{magenta}{\dagger}$ Olivier Guéant, Charles-Albert Lehalle, Joaquin Fernandez Tapia, Dealing with the Inventory Risk. A solution to the market making problem, 2011.

## Answer by lehalle (score 2)

https://quant.stackexchange.com/a/77790

It means that the quotes $\delta^a,\delta^b$ of the market maker

- cannot use future information (it is what predictable means)

- has to be finite (this is for bounded from below).

The second assumption is technical (to obtain the theorems). If the first one seems natural, do not forget that in some market making academic papers (like Kyle 87), some traders have future information. Typically the asset managers are meant to be able to understand news and other data about the companies, and it is because of that that they take position.

But in the context of this paper, that I recommend by the way ;{)}, the market maker only observes flows (that drives the intensity of being hit at the bid or ask), that reflects in its inventory, to take decisions.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.